Vol 4, No 12 (2016)

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Jagabandhu Saha IJSRE Volume 4 Issue 12 December 2016 Page 6078

Volume||4||Issue||12||December-2016||Pages-6078-6084||ISSN(e):2321-7545 Website: http://ijsae.in

DOI: http://dx.doi.org/10.18535/ijsre/v4i12.01

A Two-Way Generalisation of The Chow Test: The General Case

Author

Jagabandhu Saha

Department of Economics and Politics, Visva-Bharatri, Santiniketan, West Bengal, India.

Email: [email protected] ABSTRACT

While Chow test is applicable for testing the equality between sets of coefficients in two linear regressions, this paper attempts to construct a test procedure not only to compare the equality between sets of coefficients in two linear regressions, but also, in case they are not equal, to provide detailed informations about the inequality of the sets. Not only that, in this paper we also try to accomplish all these for not just only two linear regressions but for all possible pairs of linear regressions out of any number of given linear regressions, thus generalising the Chow test in two directions. The procedure is then illustrated through comparison of Engel curves on food for different Socioeconomic groups of Rural India, using NSS data. Key words: Chow Test, Detailed informations, All possible pairs of linear regressions.

1. INTRODUCTION:

Testing the equality between sets of coefficients in two linear regressions by Chow Test

(Chow 1960) [1] is well known. There is no problem if in the Chow Test the null hypothesis of equality between the sets of coefficients is not rejected (as in the examples in his paper). But if rejected, then, naturally, one is probed to the questions: a). at which component/s the sets differ, and b). for each of these components, between the two coefficients of the two regressions concerned, which one is larger/smaller. Chow test, however, does not provide any answer to these questions at all. These problems can be resolved with some modifications of the model (Saha and Pal 2014) [2]. Saha and Pal introduced the concept of “component wise complete comparison” (CCC)1

in order to overcome this problem. The test procedure for CCC between every two successive regressions out of any number of given successive regressions was developed. Now, this paper attempts to construct a test procedure for CCC between not only every two successive regressions out of any number of given successive regressions, but between the two linear regressions of all possible pairs of regressions out of any number of given regressions, thus providing a two-way generalisation of the Chow test. The rest of the paper can be outlined as follows. In Section 2, we put the problem in the formal terms. Section 3 is devoted for the methodology for solving the problem concerned. In Section 4 we consider a numerical example in order to illustrate the methodology while in Section 5 we present our conclusions.

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By complete comparison between any two parameters a and b we mean to decide whether a < b or a = b or a > b. By component wise complete comparison (CCC) between two vectors of parameters of the same size (a1 a2 … am) and (b1 b2 … bm) we mean

complete comparison between (a1 and b1), (a2 and b2), ... and (am and bm). By CCC between/of/for two regressions with same no.

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Jagabandhu Saha IJSRE Volume 4 Issue 12 December 2016 Page 6079 2. THE MODEL:

We consider the problem of finding test procedure for CCC between the two regressions of all possible pairs of regressions out of m given regressions as follows:

y (1) = a1(1) + a2 (1) x2 (1) + a3 (1) x3 (1) + ….. + ak (1) xk (1) + u (1), y (2) = a1(2) + a2 (2) x2 (2) + a3 (2) x3 (2) + ….. + ak (2) xk (2) + u (2), ………

y (m) = a1(m) + a2 (m) x2 (m) + a3 (m) x3 (m) + ….. + ak (m) xk (m) + u (m), 1 … (1) where, n1, n2, …, nm are the nos. of observations for these regressions. We, assume as in Chow test that the components of each of u(1), u(2), …, u(m) are iid N(0 , ). Also the vectors

u(1), u(2), …, u(m) are mutually independent.

Now, we proceed as follows. Firstly, for CCC between first and second regressions in (1), first and third regressions, …, first and m-th regressions, one requires to decide whether the differentials:

c j12 = a j2 - a j1 < 0 or = 0 or > 0, for all j = 1, 2, …, k,

………

c j1m = a jm - a j1 < 0 or = 0 or > 0, for all j = 1, 2, …, k.

Similarly, for second and third regressions, second and fourth regressions, …, second and

m-th regressions, one requires to decide whether the differentials:

………

c j2m = a jm - a j2 < 0 or = 0 or > 0, for all j = 1, 2, …, k. ………

Lastly, for (m-1)-th and m-th regressions, one requires to decide whether the differentials:

c jm-1,m = a jm - a jm-1 < 0 or = 0 or > 0, for all j = 1, 2, …, k.

A moment’s reflection shows that the desired CCC, i.e., CCC for all pairs of regressions out of the m regressions in (1), will be over when all the decisions enlisted just above, in (m-1) groups, so to say, are completed.

3. THE METHODOLOGY:

As indicated at the end of the last Section, Methodology consists of (m-1) steps as follows:

1) combine the m regressions in (1) into a single regression equation model; modify the combined model in such way that the differentials cj12, cj13, …, cj1m, for all j = 1, 2, …, k, appear as

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perform tests on the regression coefficients of this model (particularly, on the differentials concerned) and decide for each of these differentials whether it is : < 0 or = 0 or > 0. This will complete CCC for (m-1)

pairs of regressions: first and second, first and third, …, first and m-th. 2). do exactly similar as in the step 1). but with the last (m-1) regressions in (1). This will cover CCC for

(m-2) pairs of regressions: second and third, second and fourth, …, second and m-th. ……… m-1). lastly, do exactly similar as in the step 1). but with only the last two regressions in (1).This will cover CCC for one pair (the last pair) of regressions: (m-1)-th and m-th.

Thus CCC for all possible pairs of regressions is complete. r

Let us work out these steps.

1). We combine the regressions in (1) as follows:

+

….. ….. (2) where, in (2) and subsequently, ni = ni , for all i = 1, 2, …, m.

To modify the model (2) in order to get the differentials cj12, cj13, …, cj1m, for all j = 1, 2, …, k, as regression coefficients, we rewrite it as:

+

….. ….. (3)

Now, we run regression with (3) and carry out tests as prescribed above.

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+ …

….. (4)

To modify the model (4) in order to get the differentials cj23, cj24, …, cj2m, for all j = 1, 2, …, k, as regression coefficients, we rewrite it as:

+ …(5)

…… (5)

Now, we run regression with (5) and carry out tests as prescribed above. ………

m-1). We combine the last two regressions in (1) as follows:

= + …(6)

In order to get the differentials cjm-1,m, for all j = 1, 2, …, k, we rewrite (6) as:

= + …

….. (7)

Now, we run regression with (7) and carry out tests as prescribed above.

In order to present the outcomes of all possible comparisons neatly, we introduce a matrix, to be called Indicator Matrix (IM), as

imxm = ( r ij ) ,

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significantly or not and is defined as follows: r ij = 0, when i = j,

0, when i j and the two regressions concerned do

not differ from each another significantly,

1, when i j and the two regressions concerned

differ from each another significantly, where, i, j = 1, 2, ….., m2.

Evidently, IM is a symmetric matrix (mxm) with all the diagonal elements as zeros. Also, IM will be a null matrix iff all the regressions coincide.

4. ILLUSTRATION:

In the context of Engel curve for food, considering, for each of the

Socioeconomic groups ST, SC, OBC, OTHERS, ALL, regression of proportion of Monthly Per Capita Expenditure on Food (pMPCEF), proportion with respect to Monthly Per Capita

Expenditure (MPCE), on MPCE, i.e., regression of pMPCEF on MPCE, totally five regression equations are constructed as follows (m = 5). State level data on Rural India on MPCE and MPCEF for the different Socioeconomic groups mentioned are used, the source of data being NSSO Report [3]3.

Define : x(i) = MPCE of a State for the ith Socioeconomic group, for all i = 1, 2, …, 5. Y

y(i) = pMPCEF of a State for the ith Socioeconomic group, for all i = 1, 2, …, 5. Then, the regressions considered are as follows:

y (i) = a1(i) + a2 (i) x2 (i) + u (i) , ………. (8) for all i = 1, 2, …, 5,

and the task is to perform CCC between the two regressions of all pairs of regressions out of these five regressions (needless to say, 10 pairs). (The no. of observations for each regression is thirty (no. of States/UTs in India)4 and, so, we have: m = 5, n = 30, k = 2.)

According to the Methodology described above, we complete the four steps5 as follows. 1). Firstly, we run the regression (3). The decisions on the basis of the regression results come out as follows:

c112 < 0, c212 > 0, c113 = 0, c213 = 0, c114 = 0, c214 = 0, c115 = 0, c215 = 0. This in turn means that as long as the relationship of MPCE with pMPCEF is concerned, the Socioeconomic group ST differs from SC significantly (at both intercept and slope

coefficients) but not from the other groups.

2). Secondly, we run the regression similar as in Step 1). but with the last four regressions in (8). The decisions are:

c123 > 0, c223 < 0, c124 > 0, c224 < 0, c125 > 0, c225 < 0.

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Of course, IM provides limited knowledge that for every two regressions, whether they coincide or not, nothing more.

3

The said data for all India are not available in the Report. But, needless to say, the rural-urban issue is in no way relevant in the present context.

4

The no. of States/UTs, as in the Report, is 35. But there 5 for each of which some figure/s is/are missing and hence we have excluded those. Also, it may be noted that the numbers of observations for the regressions need not be the same.

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This, in terms of the relationship concerned, means that the Socioeconomic group SC differs from each of OBC, OTHERS and ALL significantly (at both intercept and slope coefficients). This is quite expected because ST differs from SC significantly but not from OBC, OTHERS and ALL. So, we can say that SC should differ significantly from all other groups.

3). Thirdly, we run the regression similar as in Step 1). but with the last three regressions in (8).

The decisions are:

c134 = 0, c234 = 0, c135 = 0, c235 = 0.

This in turn means that the Socioeconomic group OBC does not differ from OTHERS and ALL significantly which also is expected from the behaviour of ST itself.

4). Lastly, we run the regression similar as in Step 1). but with the last two regressions in (8).

The decisions are:

c145 = 0, c245 = 0.

This in turn means that the Socioeconomic group OTHERS does not differ from ALL significantly which also is expected from the behaviour of ST itself.

So, the indicator matrix for the five regressions under consideration is:

i 5x5 = ……… (9)

5. CONCLUSIONS:

The above test procedure enables one, for each and every pair of regressions possible out of the m given regressions, to perform CCC, or, in other words, to bring out the results of detailed comparison, between the two regressions in that pair.

Needless to say, this generalises the Chow test in two directions. This, further, has the following important implications.

Firstly, suppose the regressions are arranged successively in a definite order for investigating the existence of structural change. Then, obviously, if this procedure is carried out, CCC between every two successive regressions out of the given regressions arranged now successively is automatically done and hence detailed informations regarding all structural changes are obtained, if there is any such at all6.

Secondly, since by this procedure one can perform CCC between the two regressions of each of all possible pairs of regressions out of the given regressions, one can partition the set of the given regressions into, say, clusters, every cluster consisting of those regressions which satisfy the condition that the vectors of coefficients of any two of these regressions do not differ from each other significantly.

As an example, in the Illustration above, for each of the Socioeconomic groups ST, SC, OBC, OTHERS and ALL, to be denoted as S1, S2, S3, S4 and S5 respectively, we have considered regression of proportion of MPCE on food (pMPCEF) on MPCE and worked out CCC between the two regressions of each of all possible pairs of regressions out of these five regressions. From the IM given by (9), we get here two clusters as follows:

C 1 = { S1, S3, S4, S5 } and

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C 2 = { S2 }.

So, it may be concluded that in respect of the relation of MPCE with proportion of MPCE on food, the behaviour of the SC group do not match with that of the other groups while the other groups behave in the same manner.

It is be noted that no. of clusters need not always be two; it may be more than two as well as be one when and only when all the regressions coincide, i.e., the indicator matrix becomes a null matrix of appropriate order.

It may be noted that the clustering introduced here is quite different from that which is obtained by using Mahalanobis D2 Statistic [4] or using k-means Method [5] --- clustering by means of the later approaches is based on a single variable/a vector of variables while that by means of our procedure is based entirely on

relationship of variables.

It is a hunch that the procedure introduced here can be extended for the purpose of clustering on the basis of not only one relationship of variables but more than one relationship together. Also, it seems that some

significant analysis is possible to be carried out in terms of the Indicator Matrix introduced here.

REFERENCES:

1. Chow, Gregory C. (1960): Tests of Equality between Sets of Coefficients in Two Linear Regressions, Econometrica, Vol. 28, No. 3, pp. 591-605.

2. Saha, J. & Pal, M. (2014): A Modified Chow Test Approach Towards Testing Differences in The Engel Elasticities, Asian-African Journal of Economics and Econometrics, Vol.14, No.1, 2014; 57-67.

3. National Sample Survey Organisation (2007), Household Consumer Expenditure Among Socio-Economic Groups: 2004 – 2005, NSS 61st Round (July 2004 - June 2005), NSSO Report No. 514(61/1.0/7), Government of India.

4. Mahalanobis P.C. (1936): On the generalised distance in statistics, Proceedings of the National Institute of Sciences of India 2 (1): 49–55. Retrieved 2012-05-03.